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Conformal Transformations

- Neil Marks,
- DLS/CCLRC,
- Daresbury Laboratory,
- Warrington WA4 4AD,
- U.K.
- Tel (44) (0)1925 603191
- Fax (44) (0)1925 603192

Use of transformations

- This is a mathematical technique developed to

give analytical expressions for potential

and flux density distributions for (simple)

geometries. - This was the standard technique for magnet design

before finite element analysis (f.e.a.) codes

were developed. - The technique is based on transforming

geometries using functions of the complex

variable. - We start with a model with unknown field and

potential distributions and transform that to a

geometry where distributions can be analytically

defined. The known distribution is then

transformed back to the initial model using the

inverse of the first transform, giving the

required result. - Each transformation often involves an

intermediate step.

The Transformations.

- We start by defining two complex planes, Z and W

All conformal transformations preserve the

angle of intersection between two curves

except at the origin and at poles of the

transforming function. As the most suitable

ideal pole to use for the known distribution

will be a pair of parallel poles extending

to plus and minus infinity, such a

function will be necessary.

The 'Schwarz - Christoffel' Transformation

- This is one type of a conformal transform it

transforms polygons (complete or open) in the

Z plane to a straight line on the real

axis in the W plane - The transformation is given by
- dZ/dW M ( W - a1)((a1/p ) -1) (W - a2)((a2/p )

- 1) ........ - where M is an arbitrary constant to be determined.

Second transformation.

- A second transformation is then used to

translate from a T plane, where the

geometry is an infinite dipole, back into

the W plane - The two transformations are then combined

to predict the distributions in the real

magnet.

Example a dipole end.

Transformation Z to W

dZ/dW M (W 1)1/2 ( W ) -1 Z M 2(W

1)1/2 ln (W 1)1/2 -1 - ln (W 1)1/2 1

N where N is another arbitrary constant N

0 just sets the origin in Z plane for W

-1, the above expression gives Z i M p i

g/2 so M g/2p for W gt 0, Z is real.

Transformation T to W

- The T plane, an infinite dipole
- dT/dW M (W a )(a/p 1)
- for the above a 0 a 0
- so dT/dW M W -1
- T M ln(W) N
- again N 0 sets the origin in the Z plane
- for W -1, above expression gives Z Mip

ig/2 - so M g/2p, giving
- T (g/2p) ln W
- W exp(2pT/g)

Resulting equation

- We now substitute for W to give Z in terms of T
- Z (g/2p) 2 exp (2pT/g) 11/2
- ln exp(2pT/g 1 -1
- - ln exp(2pT/g 1 1
- Equipotential lines are Im (T) const
- Flux lines are Re (T) const.
- Expanding with Re and Im of T and then separating

the real and imaginary components of Z is long

and detailed see next slide.

Solution

For g 1

where

Graphical results - lines of scalar potential

Graphical results - lines of flux

Flux density B

- B -? f
- put T y if
- then Bx - ?f / ? x
- By - ? f / ? y
- Bx - j By - ? f/? x i ? f/? y
- from Cauchy-Riemann equations
- ? y / ? x ? f / ? y
- ? y / ? y - ? f/? x
- so Bx - iBy
- - ? f / ? x i? y / ? x
- i ? T / ? x
- hence
- B d T/ dZ
- ( dT/dW )( dW/dZ )

Flux density on x axis

Problems

- integration is only analytical for angles of 0

or multiples of p/2 - and a limited number of right angles
- other more complex geometries require numerical

integration - predicts distributions only for µ ? in the

steel - the technique takes no account of coils ie all

currents are at infinity.

The 'Rogowski' roll-off

- The classical end solution, developed originally

in electrostatics during the study of the end

effect for two parallel capacitor plates. The

analysis also uses the conformal transformation

method

Analysis

- dZ/dW M (W 1)/W
- Z (d/?)( 1 W ln W )
- T (1/?) ln W
- so Z (d/?)( 1 exp ?T ?T )
- if T y jf
- then in the Z plane
- x (d/?)(1 (exp py)(cos ?f) ?y)
- y (d/?)( (exp ?y)(sin ?f) ?f)
- Potential is lines of const f stream lines are

const y .

Graphical results

- Potential lines

Blown-up version

The central heavy line is for f 0.5. Rogowski

showed that this was the fastest changing line

along which the field intensity was monotonically

decreasing.

Application to magnet ends

- Conclusion Recall that a high µ steel surface is

a line of constant scalar potential. Hence, a

magnet pole end using the f 0.5 potential line

will see a monotonically decreasing flux density

normal to the steel at some point where B is

much lower, this can break to a vertical end

line.

Magnet half gap height g/2 Centre line of

gap is y 0 Equation is y g/2

(g/?) exp ((?x/g)-1)